Least-Squares Covariance Matrix Adjustment

We consider the problem of finding the smallest adjustment to a given symmetric
n by n matrix, as measured by the Euclidean or Frobenius norm, so that it
satisfies some given linear equalities and inequalities, and in addition is
positive semidefinite. This least-squares covariance adjustment problem is a
convex optimization problem, and can be efficiently solved using standard
methods when the number of variables (i.e., entries in the matrix) is modest,
say, under 1000. Since the number of variables is n(n+1)/2,
this corresponds to
a limit around n=45.
Malik [2005] studies a closely related problem, and calls
it the semidefinite least-squares problem. In this paper we formulate a dual
problem that has no matrix inequality or variables, and a number of (scalar)
variables equal to the number of equality and inequality constraints in the
original least-squares covariance adjustment problem. This dual problem allows
us to solve far larger least-squares covariance adjustment problems than would
be possible using standard methods. Assuming a modest number of constraints,
problems with n=1000 are readily solved by the dual method. The dual method
coincides with the dual method proposed by Malik when there are no inequality
constraints, and can be obtained as an extension of his dual method when there
are inequality constraints. Using the dual problem, we show that in many cases
the optimal solution is a low rank update of the original matrix. When the
original matrix has structure, such as sparsity, this observation allows us to
solve very large least-squares covariance adjustment problems.